Timeline for What happens if "equality of sets" is regarded as a defined concept instead of a primitive notion?
Current License: CC BY-SA 4.0
17 events
| when toggle format | what | by | license | comment | |
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| Jan 22 at 18:39 | audit | Reopen votes | |||
| Jan 22 at 19:47 | |||||
| Jan 13 at 8:50 | audit | Close votes | |||
| Jan 13 at 8:52 | |||||
| Jan 2 at 9:42 | vote | accept | Pedro | ||
| Dec 28, 2023 at 5:10 | answer | added | NikS | timeline score: 0 | |
| Dec 28, 2023 at 4:20 | answer | added | Greg Nisbet | timeline score: 1 | |
| Dec 27, 2023 at 23:39 | history | became hot network question | |||
| Dec 27, 2023 at 23:23 | comment | added | Alex Kruckman | @Pedro Equality is a primitive notion of first-order logic. There is another logic, called "first-order logic without equality" in which equality is not taken to be primitive. But standard set theories like ZFC are formulated in first-order logic, with equality as a primitive notion. | |
| Dec 27, 2023 at 23:06 | comment | added | Asaf Karagila♦ | @Z.A.K.: Is that really the case even for a language as simplistic as $\{\in\}$? Is that the whole thing about Leibnizian equality? | |
| Dec 27, 2023 at 18:06 | comment | added | MJD | @Z.A.K.Thanks for this and your comment below. I didn't appreciate this issue. | |
| Dec 27, 2023 at 17:27 | comment | added | Pedro | @Z.A.K. So equality is neither a primitive notion nor a defined concept, right? It is a "previous thing" whose properties (such as, for example, substitution) are already known when we state the axiom of extensionality. Is this correct? What is he technical name for what I called “previous thing”? | |
| Dec 27, 2023 at 17:01 | answer | added | Chad K | timeline score: 3 | |
| Dec 27, 2023 at 16:56 | answer | added | TurlocTheRed | timeline score: -4 | |
| Dec 27, 2023 at 16:28 | comment | added | Z. A. K. | @MJD: One cannot regard $S=T$ as a defined abbreviation for $\forall x. x\in S \leftrightarrow x\in T$ in ZF set theory. That simply does not sufficie to get the essential substitution property of equality. | |
| Dec 27, 2023 at 16:20 | comment | added | MJD | In axiomatic set theory, #2 is exactly how it is defined. The usual language of axiomatic set theory is just the language of first-order logic, plus one undefined relation “$\in$”. The notation “$S=T$” is understood as nothing more than a convenient abbreviation for “$\forall x(x\in S\iff x\in T)$”. | |
| Dec 27, 2023 at 16:08 | answer | added | ac15 | timeline score: 15 | |
| Dec 27, 2023 at 16:04 | answer | added | Lee Mosher | timeline score: 0 | |
| Dec 27, 2023 at 15:36 | history | asked | Pedro | CC BY-SA 4.0 |